# lognormal VAR

#### ajsa

##### New Member
Hi David,

Could you explain how lognormal VAR's formula is derived?

Thanks,

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi ajsa,

Sure, this alternative approach only recently finds an introduction in the FRM with Dowd's Liquidity VaR.

We typically (following Jorion) use the "arithmetic" approach;
i.e., absolute returns VaR (%) = -mean + (volatility * deviate)
where the "cutoff return" r* = mean - volatilty * deviate

instead under Dowd's "geometric" approach (i.e., log returns are normal --> prices are lognormal):
r* = LN (today's price/yesterday's price)
r* = LN(P2) - LN(P1)
LN(P2) = r* + LN(P1)
EXP(LN(P2)) = EXP(r* + LN(P1))
today's price P2 = EXP(r*) * P1
P1 - P2 = P1 - EXP(r*) * P1
(P1 - P2) = P1*(1 - EXP(r*))
source: Dowd page 62

and if today's price P2 = cutoff value (i.e., loss that corresponds to VaR) = P2*,
then VaR = P1 - P2* = P1 * (1 - EXP(r*))

where r* = mean - volatility * deviate, so that
VaR = P1 - P2* = P1 * (1 - EXP(mean - volatility * deviate))

or % "absolute" lognormal VaR = 1 - EXP(mean - volatility * deviate)
e.g., 10% mean return and 20% volatility = 1 - EXP(10% - 20% * deviate) = 20.5% @ 95% confidence
compare to "arithmetic" of -10% + (20 * 1.645) = 22.9%

Hope that helps, David

#### Stuti

##### Member
Hi David,

I am unable to understand how are we saying : "cutoff return" r* = mean - volatilty * deviate

Also,
where r* = mean - volatility * deviate, so that
VaR = P1 - P2* = P1 * (1 - EXP(mean - volatility * deviate)) - How is VaR = P1-P2?

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi
the maximum % loss that the portfolio can suffer in terms of percentage is r*=cutoff return=mean - volatilty * deviate
This essentially implies that the minimum value that the portfolio can have is P2* =P1 * (1 - EXP(r*)) where r*=cutoff return is the return at which portfolio suffers max loss.
The Var is the maximum loss that the portfolio can suffer at a given confidence level=P1 - P2*,where P2* is the minimum value that the portfolio can attain therefore P1 - P2* is the maximum loss that the portfolio can suffer which is nothing but Var.The P1 - P2* is the Value at Risk(Var).
thanks

#### NNath

##### Active Member
Hi @ShaktiRathore @David Harper CFA FRM, If we had to scale the lognormal VAR using the square root rule i.e. if we are given annual return and volatility, should we scale down the mean and volatility before calculating lognormal var or scale down the calculated log normal var. In the below question from GARP

The annual mean and volatility of a portfolio are 10% and 40%, respectively. The current value of the portfolio is GBP 1,000,000. How does the 1-year 95% VaR that is calculated using a normal distribution assumption (normal VaR) compare with the 1-year 95% VaR that is calculated using the lognormal distribution assumption (lognormal VaR)?

what if it was 1-day 95% VAR. Just calculate lognormal var ?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @NNath

Your safest approach is to scale the mean and volatility separately, because you can't scale them after they are combined. I think we identified an error in this question, see https://forum.bionicturtle.com/threads/2013-garp-exam-practice-quesiton.9110/

Here is the XLS, I just added a daily column, https://www.dropbox.com/s/r12ictk7uv8hr4y/0407-garp-2013-p2-q2.xlsx?dl=0
notice i first did:
• daily drift = 10%/250
• daily vol = 40%/sqrt(250). I don't think there is a shortcut, I hope that helps! Last edited:

#### frmqiu

##### New Member
Sorry I still have trouble in understanding the lognormal VaR calculation method. Could you confirm with me that the VaR calculation method shown on the "2016 Financial Risk Manager Examination (FRM) Part II Practice Exam" is not correct?

The annual mean and volatility of a portfolio are 12% and 30%, respectively. The current value of the portfolio is GBP 2,500,000. How does the 1-year 95% VaR that is calculated using a normal distribution assumption compare with the 1-year 95% VaR that is calculated using the lognormal distribution assumption?

[Solution Key] The lognormal VaR is calculated as follows:
Lognormal VaR(%)=0.12-exp[0.12-(1.645*0.3)]=0.56832 = 56.83%.

Do you know where this VaR formula above is coming from?

Hi @NNath

Your safest approach is to scale the mean and volatility separately, because you can't scale them after they are combined. I think we identified an error in this question, see https://forum.bionicturtle.com/threads/2013-garp-exam-practice-quesiton.9110/

Here is the XLS, I just added a daily column, https://www.dropbox.com/s/r12ictk7uv8hr4y/0407-garp-2013-p2-q2.xlsx?dl=0
notice i first did:
• daily drift = 10%/sqrt(250)
• daily vol = 40%/sqrt(250). I don't think there is a shortcut, I hope that helps! #### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @frmqiu It is a mistake in the practice paper. Per Dowd, lognormal VaR(%) = 1 - exp(µ - σ*z), which in this case is 1-exp(12% - 30%*1.645) = 31.16% or 31.16%*2,500,000 = GBP 779,122 such that the answer looks like it should be (37.35% - 31.16%)*2,500,000 = GBP 154,518 is how much higher is the normal VaR than the lognormal VaR. I hope that is helpful!

#### frmqiu

##### New Member
Thank you so much, David! I got it.

#### Serge72

##### New Member
Hi @NNath

daily drift = 10%/sqrt(250)
There is a typo. It should be daily drift = 10%/250 as it is in the spreadsheets.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Thank you @Serge72 I really appreciate the attention to detail corrected above

#### MisterDiaz

##### New Member
Looks like Lognormal Var is generally farther inside the dist than Normal Var?

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Last edited:

#### nitin3000

##### New Member
I had the same doubt as frmqiu above. It looks like the 2016 answer sheet is using a wrong formula for Log normal VAR

#### Nicole Seaman

##### Director of FRM Operations
Staff member
I had the same doubt as frmqiu above. It looks like the 2016 answer sheet is using a wrong formula for Log normal VAR
@nitin3000

I want to point out that we do not recommend using GARP's practice exams prior to 2018, as there are many errors that are going to just confuse you. I really suggest only using the 2019 and 2020 practice exams.

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